Let $R$ be a ring of subsets of a set $Ω$ and $μ: R → [0, + ∞]$ be a $\sigma$-additive pre-measure on a $R$.
The Carathéodory's extension theorem states that there exists a measure $μ′: σ(R) → [0, + ∞]$ such that $μ′$ is an extension of $μ$.
I was wondering if an extension exists when $R$ is a $\pi$ system?
The reason for considering a $\pi$ system is because a measure on a sigma algebra generated by a $\pi$ system is uniquely determined by its restriction on the $\pi$ system, if the measure is $\sigma$-finite wrt the $\pi$ system. I want to know about the existence of extension instead.
Thanks an regards!